Is a domed plane possible?
Yes, but it may be tricky to get a stack of modifiers that will work well for you modifiers. Here's an example with a simple target shape made with 2D Bezier curves getting warped in 3D space by a Lattice modifier:



The mesh created by the bezier object isn't going to play will with deformations, but the Remesh modifier will. But that won't work on something flat, so you need to use Solidify first. Give it a small thickness if you want the end result to still seem flat.

Depending on your shape, Remesh may need a larger Octree Depth, and sharpness tweaking.

Next, to fix the smooth shading around the edges, I added an EdgeSplit - the default options should work well.

Finally, the deform is done with a Lattice. If you aren't familiar with using a Lattice to deform objects, these are a little out of date, but still accurate: https://wiki.blender.org/index.php/Doc:2.6/Tutorials/Modifiers/Lattice/Shaping_a_Fork 

If you don't need the level of control the Lattice gives you, you could also try the Cast modifier (with an Empty as the center of the virtual sphere it is being cast to):